MATH
M3L5

D nagesh kumar iisc optimization methods m3l5 14

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D Nagesh Kumar, IISc Optimization Methods: M3L5 14 Finding Dual of a LP problem…contd. RHS of i th constraint constraints Cost coefficient associated with i th variable in the objective function Cost coefficient associated with j th variable in the objective function RHS of j th constraint j th variable unrestricted j th constraint with = sign i th constraint with = sign i th variable unrestricted Dual Primal Refer class notes for pictorial representation of all the operations

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D Nagesh Kumar, IISc Optimization Methods: M3L5 15 Finding Dual of a LP problem…contd. Note: Before finding its dual, all the constraints should be transformed to ‘less-than-equal-to’ or ‘equal-to’ type for maximization problem and to ‘greater-than-equal-to’ or ‘equal-to’ type for minimization problem. It can be done by multiplying with -1 both sides of the constraints, so that inequality sign gets reversed.
D Nagesh Kumar, IISc Optimization Methods: M3L5 16 Finding Dual of a LP problem: An example Primal Dual Maximize Minimize Subject to Subject to 6000 3 2 2 1 + x x 2000 2 1 - x x 4000 1 x 3 3 2 2 1 + y y 4 3 2 1 = + - y y y 1 unrestricted x 0 2 x 0 1 y 0 2 y 2 1 3 4 x x Z + = 3 2 1 4000 2000 6000 y y y Z + - = Note: Second constraint in the primal is transformed to before constructing the dual. 1 2 2000 x x - + ≤ - 0 3 y

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D Nagesh Kumar, IISc Optimization Methods: M3L5 17 Primal-Dual relationships If one problem (either primal or dual) has an optimal feasible solution, other problem also has an optimal feasible solution. The optimal objective function value is same for both primal and dual. If one problem has no solution (infeasible), the other problem is either infeasible or unbounded. If one problem is unbounded the other problem is infeasible.
D Nagesh Kumar, IISc Optimization Methods: M3L5 18 Dual Simplex Method Simplex Method verses Dual Simplex Method 1. Simplex method starts with a nonoptimal but feasible solution where as dual simplex method starts with an optimal but infeasible solution. 2. Simplex method maintains the feasibility during successive iterations where as dual simplex method maintains the optimality.

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D Nagesh Kumar, IISc Optimization Methods: M3L5 19 Dual Simplex Method: Iterative steps Steps involved in the dual simplex method are: 1. All the constraints (except those with equality (=) sign) are modified to ‘less-than-equal-to’ sign. Constraints with greater-than-equal-to’ sign are multiplied by -1 through out so that inequality sign gets reversed. Finally, all these constraints are transformed to equality sign by introducing required slack variables. 2. Modified problem, as in step one, is expressed in the form of a simplex tableau. If all the cost coefficients are positive (i.e., optimality condition is satisfied) and one or more basic variables have negative values (i.e., non-feasible solution), then dual simplex method is applicable.
D Nagesh Kumar, IISc Optimization Methods: M3L5 20 Dual Simplex Method: Iterative steps… contd.

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